Final answer:
To calculate the total cost of producing 20 units, integrate the given marginal cost function C'(q) from 0 to 20 and add the fixed cost C(0) of $700. The variable cost of producing 20 units is the integral of C'(q), which is $8666.67. Therefore, the total cost of producing 20 units is $9366.67.
Step-by-step explanation:
The student's question concerns the calculation of the total cost, fixed cost, and variable cost for producing a certain quantity of units, given a marginal cost function and an initial cost condition.
To find the total cost of producing 20 units, we need to integrate the marginal cost function C'(q) from 0 to 20 and then add the fixed cost C(0). The marginal cost function is C'(q) = q^2 - 15q + 70. The fixed cost C(0) is given as $700, which is the cost incurred at zero production, hence the vertical intercept of the total cost curve.
Step 1: Calculate the integral of C'(q):
- Integral of C'(q) = Integral of (q^2 - 15q + 70) dq from 0 to 20
- Integral of C'(q) = (1/3)q^3 - (15/2)q^2 + 70q evaluated from 0 to 20
- Integral of C'(q) = (1/3)(20)^3 - (15/2)(20)^2 + 70(20) - [(1/3)(0)^3 - (15/2)(0)^2 + 70(0)]
- Integral of C'(q) = 8000/3 - 3000 + 1400
- Integral of C'(q) = 8666.67 dollars
Step 2: Add the fixed cost to find the total cost:
- Total Cost = Integral of C'(q) + C(0)
- Total Cost = 8666.67 + 700
- Total Cost = 9366.67 dollars
Fixed Cost = $700 (This is the cost at zero production)
Variable Cost for producing 20 units = Integral of C'(q) = $8666.67 (This is the total cost minus the fixed cost for producing 20 units)
The total cost of producing 20 units is therefore $9366.67 dollars.